for small deviations from the level \(u_{0}\) , with \(|\epsilon | \ll 1\)  being a measure of this
deviation. In the moving, scaled coordinates \(\xi = \epsilon \left( x - U z \right)\)  and
\(\eta = \epsilon ^{3} z\) , with \(U^{2} = (2/q) u_{0}^{2}\) , the correction \(u_{1}\)  satisfies a KdV equation with
fifth order dispersion, or Kawahara equation, [1]}, [2]}
\(\frac{\partial u_{1}}{\partial \eta } + 3\sqrt{\frac{2}{q}}u_{1}\frac{\partial u_{1}}{\partial \xi }+ \sqrt{\frac{2}{q}}\frac{u_{0}}{4}\left( \frac{\nu }{q} - \frac{q}{4u_{0}^{2}} \right) \frac{\partial ^{3}u_{1}}{\partial \xi ^{3}}+ \sqrt{\frac{2}{q}}\frac{3\epsilon ^{2}\nu ^{2} u_{0}^{2}}{16q^{2}} \frac{\partial ^{5} u_{1}}{\partial \xi ^{5}} = 0.\) 
